One avenue of active research seeks to solve a more general problem called the Calderón problem for Riemannian manifolds [1]}, [2]}, [3]}, [4]}, [5]}. In this setting, one attempts to reconstruct an \(n\) -dimensional smooth manifold \(M\)  and metric \(g\)  from the Dirichlet-to-Neumman (DN) map \(\Lambda \)  defined on the boundary traces of harmonic differential forms, i.e., forms in the kernel of the Laplace–Beltrami operator \(\Delta \) . Given the Dirichlet boundary condition \(\phi \) , we have a unique harmonic \(u\) . The DN map is defined by \(\Lambda \phi \iota ^*(\star du)\)  which is valid for any \(k\) -form \(\phi \) . Fixing \(\phi \)  to be a 0-form then yields the classical problem discussed previously and in dimension 3, this is equivalent to the EIT problem. From complete knowledge of pairs of Dirichlet and Neumann data \((\phi , \iota ^* (\star du))\)  for forms of all grade, the geometric inverse problem is to determine the pair \((M,g)\)  up to isometry. If you would like, please see Uhlmann's article [6]} for an excellent explanation of the relationship between the EIT and Calderón problem.
